Hypothesis Testing Using Rejection Regions In Exercises 7–12, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.


Changing Jobs A researcher claims that 40% of U.S. adults would consider changing jobs. In a random sample of 50 U.S. adults, 25 say they would consider changing jobs. At α=0.10, is there enough evidence to reject the researcher’s claim?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H0) and alternative hypothesis (Ha). The claim is that 40% of U.S. adults would consider changing jobs. This translates to the null hypothesis H0: p = 0.40, where p is the population proportion. The alternative hypothesis Ha depends on the context; since the problem does not specify a direction, we assume Ha: p ≠ 0.40 (two-tailed test).
2. Step 2: Determine the critical value(s) and rejection region(s). For a significance level of α = 0.10 and a two-tailed test, divide α by 2 to find the area in each tail (α/2 = 0.05). Use a z-table or standard normal distribution to find the critical z-values corresponding to these tail areas. The rejection regions are z < -z_critical or z > z_critical.
3. Step 3: Calculate the standardized test statistic z. Use the formula z = (p̂ - p) / √(p(1-p)/n), where p̂ is the sample proportion, p is the hypothesized population proportion, and n is the sample size. Substitute the values: p̂ = 25/50 = 0.50, p = 0.40, and n = 50, then compute the z-statistic.
4. Step 4: Compare the calculated z-statistic to the critical z-values. If the z-statistic falls within the rejection regions (z < -z_critical or z > z_critical), reject the null hypothesis H0. Otherwise, fail to reject H0.
5. Step 5: Interpret the decision in the context of the original claim. If H0 is rejected, conclude that there is enough evidence to reject the researcher’s claim that 40% of U.S. adults would consider changing jobs. If H0 is not rejected, conclude that there is not enough evidence to reject the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample to reject H0 in favor of Ha.

Rejection Region

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level (α), which defines the probability of making a Type I error (rejecting H0 when it is true). For a given α, critical values are calculated, and if the test statistic falls within this region, the null hypothesis is rejected.

Standardized Test Statistic

The standardized test statistic, often denoted as z, measures how many standard deviations an observed sample statistic is from the hypothesized population parameter under the null hypothesis. It is calculated using the formula z = (observed proportion - hypothesized proportion) / standard error. This statistic is crucial for determining whether the sample provides sufficient evidence to reject the null hypothesis.